Univariate Descriptive Statistics: Central Tendency and Skewness

Measures of Central Tendency

• Mean: The arithmetic average calculated by summing all scores and dividing by the sample size ($n$).

  • Population mean formula: $\mu = \frac{\sum X}{N}$

  • Sample mean formula: $\bar{X} = \frac{\sum X}{n}$ or $M = \frac{\sum X}{n}$

• Median: The score corresponding to the midpoint or 50th percentile, at or below which $50\%$ of scores fall when ranked.

  • Position formula for ranked data: $\frac{n+1}{2}$

  • For an even number of scores, average the two middle values.

• Mode: The most frequently occurring score or category in a distribution.

Measurement Levels and Measures of Central Tendency

• Nominal: Mode only.

• Ordinal: Mode and Median.

• Interval: Mode, Median, and typically Mean.

• Ratio: Mode, Median, and Mean.

Distribution Shapes and Modality

• Unimodal: Distribution containing a single peak.

• Bimodal: Distribution with two categories/scores sharing the same highest frequency.

• Multimodal: Distribution with more than two peaks sharing the highest frequencies.

• Uniform (No Mode): Distribution where all scores or categories occur with equal frequency.

Statistical Properties of Central Tendency

• Sufficiency: Utilizes all score values in the sample during calculation (Mean is sufficient; Median and Mode are not).

• Resistance: Insensitivity to extreme scores (Median and Mode are resistant; Mean is non-resistant).

• Unbiasedness: Average sample statistic across repeated sampling equals the population parameter.

• Efficiency: Extent of variation in sample statistics over repeated sampling.

Skewness and Distribution Symmetry

• Symmetrical & Unimodal: Mean, Median, and Mode are equal.

• Negatively Skewed: Left-skewed distribution where $\text{Mode} > \text{Median} > \text{Mean}$.

• Positively Skewed: Right-skewed distribution where $\text{Mean} > \text{Median} > \text{Mode}$.

• Reporting Strategy: Report both Median and Mean when dealing with skewed distributions.